International Journal of Differential Equations
Volume 2009, Issue 1 395894
Research Article
Open Access

On Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial Conditions

Magdy A. El-Tawil, 

Corresponding Author

Magdy A. El-Tawil

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12613, Egypt cu.edu.eg

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Maha A. El-Hazmy, 

Maha A. El-Hazmy

Department of Mathematics, Girls College, Medina, Saudi Arabia

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First published: 29 September 2009
https://doi.org/10.1155/2009/395894
Academic Editor: Roger Grimshaw

Abstract

A perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.

1. Introduction

The nonlinear Schrodinger (NLS) equation is the principal equation to be analyzed and solved in many fields, see [1–5], for examples. In the last two decades, there are a lot of NLS problems depending on additive or multiplicative noise in the random case [6, 7] or a lot of solution methodologies in the deterministic case.

Wang et al. [8] obtained the exact solutions to NLS using what they called the subequation method. They got four kinds of exact solutions of
(1.1)
for which no sign to the initial or boundary conditions type is made. Xu and Zhang [9] followed the same previous technique in solving the higher-order NLS:
(1.2)
Sweilam [10] solved
(1.3)
with initial condition u(x, 0) = g(x) and boundary conditions ux(L0, t) = ux(L1, t) = 0, which gives rise to solitary solutions using variational iteration method. Zhu [11] used the extended hyperbolic auxiliary equation method in getting the exact explicit solutions to the higher-order NLS:
(1.4)
without any conditions. Sun et al. [12] solved the NLS:
(1.5)
with the initial condition ψ(x, 0) = ψ0(x) using Lie group method. By using coupled amplitude phase formulation, Porsezian and Kalithasan [13] constructed the quartic anharmonic oscillator equation from the coupled higher-order NLS. Two-dimensional grey solitons to the NLS were numerically analyzed by Sakaguchi and Higashiuchi [14]. The generalized derivative NLS was studied by Huang et al. [15] introducing a new auxiliary equation expansion method. Abou Salem and Sulem [16] studied the effective dynamics of solitons for the generalized Schrodinger equation in a random potential. El-Tawil [17] considered a nonlinear Schrodinger equation with random complex input and complex initial conditions. Colin et al. [18] considered three components of nonlinear Schrodinger equations related to the Raman amplification in a plasma. In [19], Jia-Min and Yu-Lu constructed an appropriate transformations and an extended elliptic subequation approach to find some exact solutions for variable coefficient cubic-quintic nonlinear Schrodinger equation with an external potential.

In this paper, a straight forward solution algorithm is introduced using the transformation from a complex solution to a coupled equations in two real solutions, eliminating one of the solutions to get separate independent and higher-order equations, and finally introducing a perturbative approximate solution to the system.

2. The Linear Case

Consider the nonhomogeneous linear Schrodinger equation:
(2.1)
where u(t, z) is a complex valued function which is subjected to
(2.2)
Let u(t, z) = ψ(t, z) + iϕ(t, z), ψ, ϕ: real valued functions. Substituting (2.2) in (2.1), the following coupled equations are got as follows:
(2.3)
where ψ(t, 0) = f1(t), ϕ(t, 0) = f2(t), G1(t, z) = −F1(t, z), G2(t, z) = F2(t, z), and all corresponding other I.C. and B.C. are zeros.
Eliminating one of the variables in (2.3), one can get the following independent equations:
(2.4)
where
(2.5)
Using the eigenfunction expansion technique [20], the following solutions for (2.4) are obtained:
(2.6)
where Tn(z) and τn(z) can be got through the applications of initial conditions and then solving the resultant second-order differential equations using the method of the variational parameter [21]. The final expressions can be got as follows
(2.7)
where
(2.8)
in which
(2.9)
The following conditions should also be satisfied:
(2.10)
Finally, the following solution is obtained:
(2.11)
or
(2.12)

3. The Nonlinear Case

Consider the homogeneous nonlinear Schrodinger equation:
(3.1)
where u(t, z) is a complex-valued function which is subjected to the initial and boundary conditions (2.2).

Lemma 3.1. The solution of (3.1) with the constraints (2.2) is a power series in ε if the solution exists.

Proof. At ε = 0, the following linear equation is got:

(3.2)
which has the solution, see the previous section,
(3.3)
Following Pickard approximation, (3.13) can be rewritten as
(3.4)
At n = 1, the iterative equation takes the following form:
(3.5)
which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained:
(3.6)
At n = 2, the following equation is obtained:
(3.7)
which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained:
(3.8)
Continuing like this, one can get
(3.9)
As n → ∞, the solution (if exists) can be reached as u(t, z) = lim n→∞ un(t, z). Accordingly, the solution is a power series in ε.

According to the previous lemma, one can assume the solution of (3.1) as the following:
(3.10)
Let u(t, z) = ψ(t, z) + iϕ(t, z), ψ, ϕ: real valued functions. The following coupled equations are got:
(3.11)
where ψ(t, 0) = f1(t), ϕ(t, 0) = f2(t), and all corresponding other I.C. and B.C. are zeros.
As a perturbation solution, one can assume that
(3.12)
where ψ0(t, 0) = f1(t), ϕ0(t, 0) = f2(t), and all corresponding other I.C. and B.C. are zeros.
Substituting (3.12) into (3.11) and then equating the equal powers of ε, one can get the following set of coupled equations:
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
and so on. The prototype equations to be solved are
(3.19)
where ψi(t, 0) = δi,0f1(t), ϕi(t, 0) = δi,0f2(t) and all other corresponding conditions are zeros. The nonhomogeneity functions and are functions computed from previous steps.

Following the solution algorithm described in the previous section for the linear case, the general symbolic algorithm in Figure 39 can be simulated through the use of a symbolic package, mathematica-5 is used in this paper.

3.1. The Zero-Order Approximation

In this case,
(3.20)
where
(3.21)
in which
(3.22)
where the constants and variables A01(z),  C02,  B01(z), , and B02(z) can be got by the aid of Section 2.
The absolute value of the zero-order approximation is got from
(3.23)

3.2. The First-Order Approximation

(3.24)
where
(3.25)
in which
(3.26)
where the constants and variables A11(z), B11(z), A12(z), and B12(z) can be evaluated in a similar manner as the zero-order approximation whereas and C12 = −B11(0).
The absolute value of the first-order approximation can be got using
(3.27)

3.3. The Second-Order Approximation

(3.28)
where
(3.29)
in which
(3.30)
where the constants and variables A21(z),  C22,  B21(z), , and B22(z) can be evaluated similarly as the previous approximation.
The absolute value of the second-order approximation can be got using
(3.31)

4. Case Studies

To examine the proposed solution algorithm, see Figure 39, some case studies are illustrated.

4.1. One Input Is On

Case Study 1 Taking f1(t) = 0, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 0, and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 1, 2, and 3.

Details are in the caption following the image
Figure 1
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 2
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 3
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

Case Study 2 Taking f1(t) = 0, f2(t) = 0, F1(t, z) = 0, F2(t, z) = 1 and following the solution algorithm, it has been noticed that the same results for the case study 1 are got.

Case Study 3 Taking f1(t) = 1, f2(t) = 0, F1(t, z) = 0, F2(t, z) = 0 and following the solution algorithm, the selective results for the first-zero approximation are got in Figures 4, 5, and 6.

Details are in the caption following the image
Figure 4
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 5
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 6
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the decrease of the solution level and its higher variability.

Case Study 4 Taking f1(t) = 0, f2(t) = 1, F1(t, z) = 0, F2(t, z) = 0 and following the solution algorithm, it has been noticed that the same results for the case study 3 are got:

4.2. Two Inputs Are On

Case Study 5 Taking f1(t) = 0, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 7, 8, and 9.

Details are in the caption following the image
Figure 7
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 8
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 9
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice that the solution level becomes a little bit higher than that of case study 2.

Case Study 6 Taking f1(t) = 1, f2(t) = 1, F1(t, z) = 0, F2(t, z) = 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 10, 11, and 12.

Details are in the caption following the image
Figure 10
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 11
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 12
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the little increase of the solution level than that of case studies 3 and 4.

Case Study 7 Taking f1(t) = 1, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 13, 14, and 15.

Details are in the caption following the image
Figure 13
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 14
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 15
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the small perturbations at small values of z.

4.3. Three Inputs Are On

Case Study 8 Taking f1(t) = 1, f2(t) = 1, F1(t, z) = 1, F2(t, z) = 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 16, 17, and 18.

Details are in the caption following the image
Figure 16
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 17
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 18
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the increase of the depth of the perturbations.

Case Study 9 Taking f1(t) = 1, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 19, 20, and 21.

Details are in the caption following the image
Figure 19
Open in figure viewerPowerPoint
The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 20
Open in figure viewerPowerPoint
The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 21
Open in figure viewerPowerPoint
The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice that the perturbations become smaller than that of case study 8.

4.4. Four Inputs Are On

Case Study 10 Taking the case of f1(t) = 1, f2(t) = 1, F1 = 1, F2 = 1, the following final results for the zero-order approximation are obtained in Figures 22, 23, and 24.

Details are in the caption following the image
Figure 22
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 23
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 24
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the little increase in the solution level.

4.5. Exponential Nonhomogeneity

Case Study 11 Taking the case of f1(t) = 1, f2(t) = 0, F1(t, z) = e−t, F2(t, z) = 0, the following final results for the zero-order approximation are obtained in Figures 25, 26, and 27.

Details are in the caption following the image
Figure 25
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 26
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 27
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice the low solution level and high perturbations.

4.6. Exponential Initial Condition

Case Study 12 Taking the case of f1(t) = e−t, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 0, the following final results for the zero-order approximation are obtained in Figures 28, 29, and 30.

Details are in the caption following the image
Figure 28
Open in figure viewerPowerPoint
The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 02 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 29
Open in figure viewerPowerPoint
The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 30
Open in figure viewerPowerPoint
The zero-order approximation |u(0)| at α = 1, T = 10, γ = 02 for different values of t, considering only one term in the series (M = 1).

One can notice that a higher solution level is got compared with case study 11 and less perturbations are got at small values of z.

4.7. First-Order Approximation

Case Study 13 Taking the case of f1(t) = 1, f2(t) = 0, F1(t, z) = 1, F2(t, z) = 0, the following final results for the zero and first-order approximations are obtained in Figures 31, 32, and 33.

Details are in the caption following the image
Figure 31
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The zero-order approximation of |u(0)| at α = 1, T = 10, γ = 0 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 32
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 0 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 33
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The zero-order approximation |u(0)| at α = 1, T = 10, γ = 0 for different values of t, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 34
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The first-order approximation of |u(1)| at α = 1, T = 10, γ = 0, ε = 01 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 35
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The first-order approximation of |u(1)| at α = 1, T = 10, γ = 0, ε = 05 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 36
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The first-order approximation of |u(1)| at α = 1, T = 10, γ = 0, ε = 1 with considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 37
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The first-order approximation |u(1)| at α = 1, T = 10, γ = 0, ε = 05 for different values of z, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 38
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The first-order approximation |u(1)| at α = 1, T = 10, γ = 0, ε = 05 for different values of t, considering only one term in the series (M = 1).
Details are in the caption following the image
Figure 39
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The general solution algorithm.

One can notice the oscillations of the solution level compared with case 7.

One can notice that the solution level increases with the increase of ε.

5. Conclusions

The perturbation technique introduces an approximate solution to the NLS equation with a perturbative nonlinear term for a finite time interval. Using mathematica, the difficult and huge computations problems were fronted to some extent for limited series terms. To get more improved orders, it is expected to face a problem of computation. With respect to the solution level, the effect of the nonhomogeneity is higher than the effect of the initial condition. The initial conditions also cause perturbations for the solution at small values of the space variable. The solution level increases with the increase of ε.

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